He said the man had an "underlying skin cancer condition [melanoma], which had already [become] complicated".
"He presented with the following symptoms: respiratory distress, shortness of breath, cardiac failure, decreased saturation and his temperature was above 38 degrees."
The man's family and 14 health workers - including three specialist doctors - who had been in contact with him were in quarantine and being monitored.
Mkhize confirmed that the number of positive Covid-19 cases in the country had increased by 93 since Saturday's figures were released.
Of those, 584 were in Gauteng, 310 in the Western Cape, 167 in KwaZulu-Natal and 72 in the Free State. There were also 100 "unallocated" cases.